[Paper Review] A Continuation Method for Large-Scale Modeling and Control: from ODEs to PDE, a Round Trip

Denis Nikitin, Carlos Canudas de Wit+1 more|arXiv (Cornell University)|Jan 25, 2021

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TL;DR

The paper introduces a continuation method that converts spatially distributed ODEs into PDEs, enabling analysis and control, with nonlinear extensions and multiple applications including Hilbert’s 6th problem and multiagent control.

ABSTRACT

In this paper we present a continuation method which transforms spatially distributed ODE systems into continuous PDE. We show that this continuation can be performed both for linear and nonlinear systems, including multidimensional, space- and time-varying systems. When applied to a large-scale network, the continuation provides a PDE describing evolution of continuous state approximation that respects the spatial structure of the original ODE. Our method is illustrated by multiple examples including transport equations, Kuramoto equations and heat diffusion equations. As a main example, we perform the continuation of a Newtonian system of interacting particles and obtain the Euler equations for compressible fluids, thereby providing an original alternative solution to Hilbert's 6th problem. Finally, we leverage our derivation of the Euler equations to control multiagent systems, designing a nonlinear control algorithm for robot formation based on its continuous approximation.

Motivation & Objective

• Motivate the need to replace large-scale spatially distributed ODEs with compact PDE representations while preserving spatial structure.
• Develop a general continuation framework that inverts discretization to obtain PDEs from ODEs for linear, nonlinear, and space- and time-varying systems.
• Establish validity and convergence results linking the PDE approximation to the original ODE spectrum.
• Demonstrate the approach on classical and novel examples, including particle systems leading to Euler equations and formation control for robots.

Proposed method

• Define a continuation procedure that replaces finite-difference discretizations with PDE representations for linear spatially invariant ODEs on a 1D grid.
• Use Taylor expansions and a matrix-based formalism to map ODE terms to PDE derivatives up to a chosen order d (with d+1 representing the PDE’s derivative order).
• Derive conditions for a valid continuation (Theorem 1: d+1 ≥ N) ensuring the discretization of the PDE recovers the original ODE.
• Analyze spectral convergence (Theorem 2) showing PDE spectrum converges to the ODE spectrum as d → ∞, with notes on nonuniform convergence and stability caveats.
• Extend the framework to nonlinear systems via computational graphs, introducing similar subgraphs and a recursive continuation to obtain nonlinear PDEs.
• Expand the method to multidimensional, space-dependent, and boundary-influenced systems, and outline how to derive Euler-like PDEs from particle dynamics and apply PDE-based control to multiagent formations.

Experimental results

Research questions

• RQ1 How can a linear spatially distributed ODE be faithfully transformed into a PDE that preserves the original spatial structure?
• RQ2 What are the accuracy and convergence guarantees when performing continuation from ODEs to PDEs, particularly in spectral terms?
• RQ3 How can the continuation framework be extended to nonlinear, space- and time-varying, and multi-dimensional systems with boundaries?
• RQ4 Can the derived PDE representations inform control design for large-scale multiagent systems, and can the continuous control be discretized back to the original agents?
• RQ5 What are concrete demonstrations (e.g., transport, Kuramoto, heat diffusion, and Euler equations from particle systems) that validate the continuation approach?

Key findings

• The PDE obtained by continuation converges spectrally to the original ODE as the continuation order increases (Theorem 2).
• A valid continuation requires the PDE derivative order satisfy d+1 ≥ N, ensuring discretization recovers the original ODE (Theorem 1).
• The framework can reproduce classical PDEs from particle dynamics, such as deriving Euler equations for compressible fluids from Newtonian interacting-particle models.
• Nonlinear dynamics can be handled using computational-graph representations, enabling continuation to nonlinear PDEs.
• The method provides a pathway to control design by deriving a PDE representation and then discretizing the designed continuous control back to the original agents (framework illustrated in Fig. 1).
• Illustrative examples include transport, Kuramoto-type interactions, and heat diffusion, as well as a Newtonian-to-Euler derivation and formation-control applications.